R70-33 Fuzzy Events Realized by Finite Probabilistic Automata

Let ∑ be a finite alphabet and ∑* the set of all finite words (sequences of symbols) over ∑. A fuzzy event f is a mapping from ∑* into [0, 1] and is called probabilistic if it is induced by a probabilistic automaton.¹Some operations are defined on fuzzy events f, g, r: for x∈∑* (f ∨ g)(x) =Max (f(x), g(x)); (f ∧ g)(x)= min (f(x), g(x)); f(x)= 1-f(x); f^T(x) =f(x^T) where x^T= σ<inf>k</inf>⋯ σ<inf>1</inf>if x = σ<inf>1</inf>⋯ σ<inf>k</inf>; [f, g:r](x)= f(x)r(x)+g(x)r̄(x), etc., and the closure of the events with regard to those operations are studied. Typical results: probabilistic events are closed under the bar operation, the bracket operation, and the "T" operation, but are not closed under the "∨" or "∧" operations (except for some restricted cases). Of particular interest is the result about the T-closure, as this problem has been open for a couple of years.